# Cyclic vector

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Let be an endomorphism of a finite-dimensional vector space . A cyclic vector for is a vector such that form a basis for , i.e. such that the pair is completely reachable (see also Pole assignment problem; Majorization ordering; System of subvarieties; Frobenius matrix).

A vector in an (infinite-dimensional) Banach space or Hilbert space with an operator on it is said to be cyclic if the linear combinations of the vectors , , form a dense subspace, [a1].

More generally, let be a subalgebra of , the algebra of bounded operators on a Hilbert space . Then is cyclic if is dense in , [a2], [a5].

If is a unitary representation of a (locally compact) group in , then is called cyclic if the linear combinations of the , , form a dense set, [a3], [a4]. For the connection between positive-definite functions on and the cyclic representations (i.e., representations that admit a cyclic vector), see Positive-definite function on a group. An irreducible representation is cyclic with respect to every non-zero vector.

How to Cite This Entry:
Cyclic vector. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cyclic_vector&oldid=18561
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article